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Chapter 5: Problem 38

Combine like terms. $$ z-z-z+z $$

Short Answer

Expert verified
The simplified expression is 0.

Step by step solution

01

Identify Like Terms

Look at the given expression: z - z - z + z The like terms here are all the instances of the variable z.
02

Group Like Terms Together

Group all the z terms together. In this expression, all terms are already grouped together as they are all z .
03

Combine Like Terms

Now, add or subtract the coefficients of the like terms. The expression has: z - z - z + z = 1z - 1z - 1z + 1z Combine these terms by performing the arithmetic operations: 1 - 1 - 1 + 1 = 0 Therefore, z - z - z + z = 0 ,

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Like Terms
In an algebraic expression, 'like terms' are terms that have the same variables raised to the same power. This means they can be combined together. For example, in the expression, \( z - z - z + z \), all the terms are 'like terms' because they all contain the variable \( z \) with no exponents. This property allows them to be grouped and simplified easily. Recognizing like terms is the first step towards combining them effectively in any algebraic expression.
Coefficients
Coefficients are the numerical part of terms in an algebraic expression. They tell you how many times the variable is being multiplied. For instance, in the term \( 1z \), the number 1 is the coefficient. When combining like terms, you focus on adding or subtracting the coefficients while keeping the variable part the same. In the example \( z - z - z + z \), each 'z' has a coefficient of 1, whether it鈥檚 explicitly written out as \( 1z \) or not. Hence, the process of combining involves dealing with the coefficients: \( 1 - 1 - 1 + 1 \), which simplifies to 0.
Arithmetic Operations
Arithmetic operations such as addition, subtraction, multiplication, and division are fundamental for combining like terms. When you combine like terms, you perform arithmetic operations on their coefficients. In the expression \( z - z - z + z \), you need to handle the subtraction and addition of the coefficients of \( z \) terms. This can be broken down as: \( 1 - 1 - 1 + 1 = 0 \). Here, you first perform \( 1 - 1 \), resulting in 0, then \( 0 - 1 \) which is -1, and finally \( -1 + 1 \) yields 0. Understanding these basic operations is crucial for simplifying expressions correctly.
Variable Manipulation
Variable manipulation involves handling variables in algebraic expressions effectively to simplify or solve them. While combining like terms, the variables themselves are preserved, and only their coefficients undergo arithmetic operations. For the expression \( z - z - z + z \), note how the variable 'z' remains unchanged throughout the process. Instead, we manipulate the coefficients (adding and subtracting them) to achieve the final simplified result. This consolidated approach of variable manipulation ensures that the expressions are simplified accurately without altering the inherent variables.

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